In a skillet over medium-high heat, combine the butter and olive oil, allowing the butter to melt completely. Introduce the sliced mushrooms and sauté leisurely until they reach a golden hue, approximately 4 to 5 minutes, stirring occasionally.
Once the mushrooms are nearly browned, incorporate the minced garlic along with a pinch of salt and pepper. Continue cooking for around 1 minute, or until the garlic becomes fragrant and lightly golden.
Pour in the wine; it should sizzle upon contact. Stir briskly, scraping any browned bits from the base of the pan, and allow the liquid to mostly evaporate within a minute.
Add the broth, cream, and finely grated parmesan to the pan. Stir well, then reduce the heat to medium, maintaining a gentle simmer. Avoid a vigorous boil to prevent the cream from curdling.
Let the sauce simmer for 2 to 3 minutes, stirring occasionally, until it achieves a slightly thickened consistency—rich but not overly dense.
Fold in the thyme leaves, adjusting seasoning with additional salt and pepper as needed. Remove from heat.
Serve generously over grilled steak, poultry, or pork. Alternatively, use it to coat pasta, fill baked potatoes, or accompany vegetables and omelettes.